Separability of Context-Free Languages by Piecewise Testable - PowerPoint PPT Presentation

Separability of Context-Free Languages by Piecewise Testable Languages Wojciech Czerwi ń ski Wim Martens

Separability

Separability K L

Separability K L S

Separability K L S S separates K and L

Separability K L S S separates K and L K and L are separable by family F if some S from F separates them

Problem

Problem Given : context-free grammars for languages K and L

Problem Given : context-free grammars for languages K and L Question : are K and L separable by piecewise testable languages (PTL)?

Problem Given : context-free grammars for languages K and L Question : are K and L separable by piecewise testable languages (PTL)? piece language

Problem Given : context-free grammars for languages K and L Question : are K and L separable by piecewise testable languages (PTL)? Σ * a 1 Σ * a 2 Σ * ... Σ * a n Σ * piece language

Problem Given : context-free grammars for languages K and L Question : are K and L separable by piecewise testable languages (PTL)? Σ * a 1 Σ * a 2 Σ * ... Σ * a n Σ * piece language piecewise testable language

Problem Given : context-free grammars for languages K and L Question : are K and L separable by piecewise testable languages (PTL)? Σ * a 1 Σ * a 2 Σ * ... Σ * a n Σ * piece language piecewise testable language bool. comb. of pieces

What is known? Separability of CFL by

What is known? Separability of CFL by • CFL - undecidable (intersection problem)

What is known? Separability of CFL by • CFL - undecidable (intersection problem) • regular languages - undecidable

What is known? Separability of CFL by • CFL - undecidable (intersection problem) • regular languages - undecidable • any family containing (reverse)-definite languages - undecidable

Definite languages

Definite languages reverse definite language = finite union of w Σ *

Definite languages reverse definite language = finite union of w Σ * any logic L

Definite languages reverse definite language = finite union of w Σ * any logic L • able to express n-th letter equals a

Definite languages reverse definite language = finite union of w Σ * any logic L • able to express n-th letter equals a • closed under boolean combinations

Definite languages reverse definite language = finite union of w Σ * any logic L • able to express n-th letter equals a • closed under boolean combinations describes all reverse definite languages

Our main result

Our main result Theorem: Separability of context free languages by piecewise testable languages is decidable

Our main message

Our main message • something nontrivial possible for separability of CFL

Our main message • something nontrivial possible for separability of CFL • no algebra needed

Our main message • something nontrivial possible for separability of CFL • no algebra needed • piecewise testable languages are special

Generalization

Generalization The same construction works for separating:

Generalization The same construction works for separating: • languages of Petri Nets

Generalization The same construction works for separating: • languages of Petri Nets • languages of Lossy Counter Machines (?)

Generalization The same construction works for separating: • languages of Petri Nets • languages of Lossy Counter Machines (?) • every class of well-behaving languages

Thank you!

Proof (sketch)

Proof (sketch) Two semi-procedures

Proof (sketch) Two semi-procedures One tries to show separability

Proof (sketch) Two semi-procedures One tries to show One tries to show separability non-separability

Proof (sketch) Two semi-procedures One tries to show One tries to show separability non-separability Enumerates all piecewise testable languages and test them

Proof (sketch) Two semi-procedures One tries to show One tries to show separability non-separability Enumerates all piecewise Enumerates all patterns testable languages and test them and test them

Patterns

Patterns Pattern p over Σ consists of:

Patterns Pattern p over Σ consists of: words w 0 , w 1 , ..., w n in Σ *

Patterns Pattern p over Σ consists of: words w 0 , w 1 , ..., w n in Σ * subalphabets B 1 , ..., B n of Σ

Patterns Pattern p over Σ consists of: words w 0 , w 1 , ..., w n in Σ * subalphabets B 1 , ..., B n of Σ B ⊗ = words from B * that contain all the letters from B

Patterns Pattern p over Σ consists of: words w 0 , w 1 , ..., w n in Σ * subalphabets B 1 , ..., B n of Σ B ⊗ = words from B * that contain all the letters from B Pattern p fits to a language L if for all k ≥ 0 intersection of L and w 0 (B 1 ⊗ ) k w 1 ... w n-1 (B n ⊗ ) k w n is nonempty

Patterns and separability

Patterns and separability Theorem (van Rooijen, Zeitoun `13): Languages K and L are non-separable by PTL if and only if there exists a pattern p, that fits to both to K and L